[FOM] 461: Reflections on Vienna Meeting

Robert Solovay solovay at gmail.com
Sun May 15 21:12:46 EDT 2011


Some perhaps obvious comments (which I prefer to make off list).

1) Usually (I don't claim always!) one can compress applications of
category theory into ZFC by "cheap tricks". What I have in mind is
instead of working with the full category (of sets, topological
spaces, chain complexes, etc.) by those in some V_kappa which (a) has
suitably high cofinality and (b) is a modl of Sigma_n replacement for
some large n.

2) My personal inclination is not to be fetishistic about working
**exactly*8 in ZFC but to follow Grothendieck's route and assume that
our universe is a V_kappa for some inaccessible kappa in some larger
universe of ZFC..

3) I think this approach (2) will cover all honest uses of Category
theory. Of course, people like Lawvere, who want there to be a
"category of all categories" have from time to time formulated systems
that are downright inconsistent. (As did Martin-Lof in a slightly
different context.)

To sum up my personal position, though I am an enthusiastic platonist,
I don't think there is anything magical about ZFC. It's just one
waystation along a long long road.

-- Bob

On Thu, May 12, 2011 at 3:17 PM, Rob Arthan <rda at lemma-one.com> wrote:
> On 12 May 2011, at 02:24, Harvey Friedman wrote:
>> 16. In particular, this matter is well worth investigating. I suspect that it is simply some fancy mathematicians (purposefully) unfamiliar with routine formalization. On the other hand, perhaps there is some new way of going beyond ZFC in some interesting sense, not known to people like me?
> It would be very helpful to know what these routine formalizations of subjects like algebraic topology that make routine use of functors between large categories look like. As a simple example, how do I go about formalising the method of acyclic models in ZFC?
> Regards,
> Rob.
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